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For a site $\mathcal{C}$, an $n$-cosheaf is a functor $\mathcal{F}:\mathcal{C}\rightarrow $n$\text{-}\mathfrak{Cat}$ such that for any cover $(U_i\rightarrow U)_i$, we have $$\prod_{i,j}\mathcal{F}(U_ …

For any groups $G,H$, we can define the category, in fact a groupoid, $$\underline{\text{Hom}}(G,H)$$ whose objects are group morphisms $G\to H$ and morphisms $(f:G\to H)\to (g:G\to H)$ are elements o …

Deligne's theorem states that a coherent topos has enough points, i.e. that we can prove that a morphism of sheaves on a "nice" site is an isomorphism by showing that the induced morphism on stalks ar …

As far as I understand, there are several ways of defining $\infty$-categories. One of them is to think of $\infty$-cateogries as $top$-enriched categories. Hence we can think of $\infty$-groupoids as …

I was wondering what a good source for the properties (or even the existence) of the abelianisation of a (2-) groupoid would be? A naive construction would certainly be to abelianise the automorphisms …

Consider two dg-algebras $A,B$ and their respective derived categories $D(A),D(B)$. A natural way to give a covariant functor is to take an $(A,B)$-bimodule $X$ and to tensor with it, that is $$D(A)\t …

Maybe there is no good answer to this, but I'm trying to get a feel for what the $K$-theory of a (permutative or symmetric monoidal $\infty$-)category computes. In algebraic $K$-theory, we have explic …

If we have a category $\mathcal{C}$, then we can see it as an $\infty$-category. Furthermore, we can truncate and $\infty$-category $\mathcal{X}$ to get a category $\mathcal{X}_{\leq 1}$. My question …

Let $\mathcal{C}$ be a site and $f:\mathcal{F}\to \mathcal{G}$ a morphism of $2$-sheaves. According to https://mathoverflow.net/q/307366, this is an epimorphism if and only if it is almost surjective …

Let $\mathcal{F}:\text{Ét}(\text{Spec}(k))^{\mathrm{op}}\rightarrow \text{Set}$ be an étale presheaf. The étale sheaf condition for the cover $\text{Spec}(l)\rightarrow \text{Spec}(k)$ is $$\mathcal{F …

By https://arxiv.org/abs/1406.4419 (The fundamental groupoid as a terminal costack, Ilia Pirashvili), we know that for a topological space $X$, the $2$-functor $$\text{Top}(X)\rightarrow \text{Gpd}, \ …